• Instructor: Anton Strezhnev

• Logistics:

  • Lectures Mon/Wed - 9:30am - 10:45pm;
  • 4 Problem Sets (~ 2 weeks)
  • Midterm (~ 1 week)
  • In-person final (TBA)
  • Office hours: Just drop in!

• What is this course about?

  • Defining statistical models via their data-generating process
  • Estimating model parameters and conducting inference
  • Interpreting model output and evaluating model quality

• Give you the tools you need to understand descriptive inference via statistical models and comment on other researchers’ work.
• Equip you with an understanding of the fundamentals of likelihood and Bayesian inference to enable you to learn new models that build on these principles.
• Connect these principles to the particular research questions that you want to answer.
• Teach you how to program and implement estimators by yourself!

• Lectures
  • Topics organized by week
  • Lectures are the “course notes” – readings are the reference manuals.
• Readings
  • Mix of textbooks and papers
  • All readings available digitally on Canvas

• Problem sets (25% of your grade)
  • Meant as a check on your understanding of the material and a way of communicating with me about the course
  • Collaboration is strongly encouraged – you should ask and answer questions on our Ed discussion board.
  • Graded holistically on a plus/check/minus system.

• Midterm and Final exam (25% and 40% of your grade)
  • The midterm exam will be structured like the problem sets with two main differences:
    • You have about 1 week to complete them instead of 2
    • You may not collaborate with one another
  • The final exam will be a written in-person exam
    • Slightly more theory heavy, but some questions will require you to analyze code + output.
• Participation (10% of your grade)
  • It is important that you actively engage with lecture and section – ask and answer questions.
  • Do the reading!
  • Participating on the discussion board counts towards this as well.

• Problem Set 1: Assigned September 9, Due September 22
• Problem Set 2: Assigned September 29, Due October 13
• Midterm Exam Assigned October 14, Due October 20 (1 week)
• Problem Set 3: Assigned October 28, Due November 10
• Problem Set 4: Assigned November 11, Due December 1 (Extra time due to Thanksgiving)
• Final Exam: TBA (whenever/wherever I can book a room)

• Overall: An interest in learning and willingness to ask questions.

• Assume a background in intro probability and statistics (1st year sequence)

  • You should be comfortable thinking about basic estimands/estimators + their properties
  • You should be able to interpret a confidence interval for (e.g.) a difference-in-means.

• Some prior knowledge of causal inference helpful but not critical

  • We’ll be connecting predictive models to causal estimands
  • Ideally should be familiar with the potential outcomes framework

• You should also be familiar with linear regression

  • \(\hat{\beta} = (\mathbf{X}^{\prime}\mathbf{X})^{-1}\mathbf{X}^{\prime}Y\) should be a familiar expression
  • You should know under what conditions it’s unbiased for \(\mathbb{E}[Y|X]\), and under what conditions it’s efficient.

• If you want some review, check out chapters 1-6 of “Regression and Other Stories”

• Understanding the behavior and properties of random variables is at the core of statistical theory.
• (Simply put) a random variable \(X\) is a mapping from a sample space to the real number line
  • Random variables have a distribution (which we may or may not assume we know) defined by the cumulative distribution function (CDF)
  \[F(x) = Pr(X \le x)\]

• Discrete random variables take on a countable number of values (e.g. Bernoulli r.v. can take on 0 or 1) and have a probability mass function (PMF)

  \[p(x) = Pr(X = x)\]

• Continuous random variables take on an uncountable number of values (e.g. the Normal distribution on \((-\infty, \infty)\)).

  • No PMF, but have a density function (PDF) that integrates to a probability

  \[Pr(X \in \mathcal{A}) = \int_{\mathcal{A}} f(x)dx\]

Remember: PMFs (and PDFs) sum (integrate) to \(1\) over the support of the random variable.

• One important property of a random variable is its expectation \(\mathbb{E}[X]\). We’ll often make assumptions about the expectation of an R.V. while remaining agnostic about its true distribution.

  • The expectation is a weighted average. For a discrete r.v. \(X\), we sum over the support of the random variable \(\mathcal{X}\).

  \[\mathbb{E}[X] = \sum_{x \in \mathcal{X}} x Pr(X = x)\]

• For continuous r.v. we have an integral

  \[\mathbb{E}[X] = \int_{x \mathcal{X}} x f(x) dx\]

• Fun fact: we can get the expectation of any function of \(g(X)\) just by plugging it into the integral

  \[\mathbb{E}[g(X)] = \int_{x \mathcal{X}} g(x) f(x) dx\]

• You’ll need to know some essential properties of expectations to simplify certain problems

• Most important. Linearity. For any two random variables \(X\) and \(Y\) and constants \(a\) and \(b\)

  \[\mathbb{E}[aX + bY] = a\mathbb{E}[X] + b\mathbb{E}[Y]\]

• Note that for any generic function \(g()\), \(\mathbb{E}[g(X)] \neq g(\mathbb{E}[X])\). If \(g()\) is convex, by Jensen’s inequality \(\mathbb{E}[g(X)] \ge g(\mathbb{E}[X])\)

• For a binary r.v. \(X \in \{0, 1\}\), it’s helpful to remember the “fundamental bridge” between expectations and probability

  \[\mathbb{E}[X] = Pr(X = 1)\]

• We also care about the spread of a random variable - how far is the average draw of \(X\) from its mean \(\mathbb{E}[X]\).One measure of this is the variance.

  \[Var(X) = \mathbb{E}[(X - \mathbb{E}[X])^2]\]

• Also written as

\[Var(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2\]

• Note that the square is a convex function. Which means that by Jensen’s inequality \(\mathbb{E}[X^2] \ge \mathbb{E}[X]^2\). Variances cannot be negative!
• We also can define a covariance between two variables (does \(X\) take high values when \(Y\) takes high values?)

\[Cov(X, Y) = E\left[(X - \mathbb{E}[X])(Y - \mathbb{E}[Y])\right] = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]\]

• Variances also have some useful properties.
• For a constant \(a\)

\[Var(aX) = a^2Var(X)\] - For any two random variables \(X\) and \(Y\)

\[Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)\] \[Var(X - Y) = Var(X) + Var(Y) - 2Cov(X,Y)\]

• For independent random variables \(X\) and \(Y\)

\[Var(X + Y) = Var(X) + Var(Y)\] \[Var(X - Y) = Var(X) + Var(Y)\]

• Key concept - Dependence and independence. If two variables are independent, the distribution of one does not change conditional on the other. We’ll write this using the \(\perp \!\!\! \perp\) notation.

  • \(Y_i \perp \!\!\! \perp D_i\) implies

  \[f(Y_i | D_i = 1) = f(Y_i| D_i = 0) = f(Y_i)\]

• Two variables can be conditionally independent in that they are independent only when conditioning on a third variable. For example, we can have \(Y_i \cancel{\perp \!\!\! \perp} D_i\) but \(Y_i \perp \!\!\! \perp D_i | X_i\). This implies

  \[f(Y_i| D_i = 1, X_i = x) = f(Y_i| D_i = 0, X_i = x) = f(Y_i | X_i =x)\]

• Remember: Conditional independence does not imply independence or vice-versa!

• A central object of interest in statistics is the conditional expectation function (CEF) \(\mathbb{E}[Y | X]\).

  • Given a particular value of \(X\), what is the expectation of \(Y\)?
  • The CEF is a function of \(X\).

• All the usual properties of expectations apply to conditional expectations.

  • We also will often make use of the law of total expectation

  \[\mathbb{E}[Y] = \mathbb{E}[\mathbb{E}[Y|X]]\]

• Easiest to think about this in terms of discrete r.v.s

  \[\mathbb{E}[Y] = \sum_{x \in \mathcal{X}} \mathbb{E}[Y | X = x] Pr(X = x)\]

• One critical use of statistical theory is understanding how to learn about things we don’t observe using things that we do observe. We call this estimation.
  • e.g. What is the share of voters in Wisconsin who will turn out in the 2026 election?
  • What is the share of voters who turn out among those assigned to receive a GOTV phone call?
• Estimand: The unobserved quantity that we want to learn about. Often denoted via a greek letter (e.g. \(\mu\), \(\pi\))
  • Often a “population” characteristic that we want to learn about via a sample.
    • Although recall causal estimands can’t be fully observed even in a finite sample!
  • Important to define your estimand well. (Lundberg, Johnson and Stewart, 2022)

• Estimator: The function of random variables that we will use to try to estimate the quantity of interest. Often denoted with a hat on the parameter of interest (e.g. \(\hat{\mu}\), \(\hat{\pi}\))
  • Why are the variables random?
    • Classic inference: We have a random sample from the population – if we took another sample, we would obtain a different realization of our estimator.
    • Randomization inference: We have a randomly assigned treatment – if we were to re-run the experiment, we would observe a different treatment/control allocation.
• Estimate: A single realization of our estimator (e.g. 0.3, 9.535)
  • We often report both point estimates (“best guess”) and interval estimates (e.g. confidence intervals).
  • Careful not to confuse properties of estimators with properties of the estimates themselves.

• The classic estimation problem in statistics is to estimate some unknown population mean \(\mu\) from an i.i.d. sample of \(n\) observations \(Y_1, Y_2, \dotsc, Y_n\).
  • We assume that each \(Y_i\) is a draw from the target population with mean \(\mu\). (identically distributed) – therefore \(\mathbb{E}[Y_i] = \mu\)
  • We’ll also assume that knowing \(Y_i\) tells us nothing about any other \(Y_j\) \(Y_i \perp \!\!\! \perp Y_j\) (independently distributed) – this implies \(Cov(Y_i, Y_j) = 0\)
• Our estimand: \(\mu\)
• Our estimator: The sample mean \(\hat{\mu} = \bar{Y} = \frac{1}{n}\sum_{i=1}^n Y_i\)
• Our estimate: A particular realization of that estimator based on our observed sample (e.g. \(0.4\))

• Note that our estimator is a random variable – it’s a function of \(Y_i\)s which are random variables.
  • Therefore it has an expectation \(\mathbb{E}[\hat{\mu}]\) (assuming \(Y_i\) has an expectation)
  • It has a variance \(Var(\hat{\mu})\) (again, under regularity conditions)
  • It has a distribution (which we may or may not know).

• How do we know if we’ve picked a good estimator? Will it be close to the truth? Will it besystematically higher or lower than the target?

• We want to derive some of its properties

  • Bias: \(\mathbb{E}\hat{\mu}] - \mu\)
  • Variance: \(Var(\hat{\mu})\)
  • Consistency: Does \(\hat{\mu}\) converge in probability to \(\mu\) as \(n\) goes to infinity?
  • Asymptotic distribution: Is the sampling distribution of \(\hat{\mu}\) well approximated by a known distribution?

• Is the expectation of \(\hat{\mu}\) equal to \(\mu\)?

  \[\mathbb{E}[\hat{\mu}] = \mathbb{E}\left[\frac{1}{n}\sum_{i=1}^n Y_i\right] = \frac{1}{n}E\left[\sum_{i=1}^n Y_i\right]\]

• Next we use linearity of expectations

  \[\frac{1}{n}\mathbb{E}\left[\sum_{i=1}^n Y_i\right] = \frac{1}{n}\sum_{i=1}^n \mathbb{E}\left[Y_i\right]\]

• Finally, under our i.i.d. assumption

  \[\frac{1}{n}\sum_{i=1}^n \mathbb{E}\left[Y_i\right] = \frac{1}{n}\sum_{i=1}^n \mu = \frac{n \mu}{n} = \mu\]

• Therefore, the bias, \(\text{Bias}(\hat{\mu}) = \mathbb{E}[\hat{\mu}] - \mu = 0\)

• What is the variance of \(\hat{\mu}\)? Again, start by pulling out the constant.

  \[Var(\hat{\mu}) = Var\left[\frac{1}{n}\sum_{i=1}^n Y_i\right] = \frac{1}{n^2}Var\left[\sum_{i=1}^n Y_i\right]\]

• We can further simplify by using our i.i.d. assumption. The variance of a sum of i.i.d. random variables is the sum of the variances.

  \[\frac{1}{n^2}Var\left[\sum_{i=1}^n Y_i\right] = \frac{1}{n^2}\sum_{i=1}^n Var\left[Y_i\right]\]

• “identically distributed”

  \[\frac{1}{n^2}\sum_{i=1}^n Var\left[Y_i\right] = \frac{1}{n^2}\sum_{i=1}^n \sigma^2 = \frac{n\sigma^2}{n^2} = \frac{\sigma^2}{n}\]

• Therefore, the variance is \(\frac{\sigma^2}{n}\)

• Rather than just estimating a population mean, we are more typically interested in some population conditional expectation \(\mathbb{E}[Y|X]\)
  • \(Y_i\): Outcome/response/dependent variable
  • \(X_i\): Vector of regressor/independent variables
• “How does the expected value of \(Y\) differ across different values of \(X\)?”
• Suppose we observe \(N\) paired observations of \(\{Y_i, X_i\}\).
  • How do we construct a “good” estimator of \(\mathbb{E}[Y|X]\)?
  • What assumptions do we have to make to get…consistency…unbiasedness…efficiency?

• Consider the ordinary least squares estimator \(\hat{\beta}\) which solves the minimization problem:

  \[\hat{\beta} = \argmin_b \ \sum_{i=1}^N (Y_i - X_ib)^2\]

• We can do some algebra and find a closed form solution for this optimization problem

  \[\hat{\beta} = (\mathbf{X}^{\prime}\mathbf{X})^{-1}(\mathbf{X}^{\prime}Y)\]

• Assumption 1: Linearity

  \[Y = \mathbf{X}\beta + \epsilon\]

• Assumption 2: Strict exogeneity of the errors

  \[\mathbb{E}[\epsilon | \mathbf{X}] = 0\]

• These two imply:

  • Linear CEF

  \[\mathbb{E}[Y|\mathbf{X}] = \mathbf{X}\beta = \beta_0 + \beta_1X_{1} + \beta_2X_{2} + \dotsc \beta_kX_{k}\]

• Best case: Our CEF is truly linear (by luck or we have a saturated model)

• Usual case: We’re at least consistent for the best linear approximation to the CEF

• Assumption 3: No perfect collinearity

  • \(\mathbf{X}^{\prime}\mathbf{X}\) is invertible
  • \(\mathbf{X}\) has full column rank

• This assumption is needed for identifiability – otherwise no unique solution to the least squares minimization problem exists!

• Fails when one column can be written as a linear combination of the others

  • Or when there are more regressors than observations \(k > n\)

• Lastly, by law of total expectation

  \[\mathbb{E}[\hat{\beta}] = \mathbb{E}[\mathbb{E}[\hat{\beta}|\mathbf{X}]]\]

• Therefore

  \[\mathbb{E}[\hat{\beta}] = \mathbb{E}[\beta] = \beta\]

• Consistency requires us to show the convergence of \((\mathbf{X}^{\prime}\mathbf{X})^{-1}(\mathbf{X}^{\prime}\epsilon)\) to \(0\) in probability as \(N \to \infty\).

  • This actually requires weaker assumptions: \(\mathbb{E}[\mathbf{X}^{\prime}\epsilon] = 0\) but not necessarily \(\mathbb{E}[\epsilon | \mathbf{X}] = 0\).

• But what have we not assumed?

  • Anything about the distribution of the errors!

• Good news! We can relax homoskedasticity (but still keep no correlation) and do inference on the variance of \(\hat{\beta}\)

  \[Var(\epsilon | \mathbf{X}) = \begin{bmatrix} \sigma^2_1 & 0 & \cdots & 0\\ 0 & \sigma^2_2 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \sigma^2_n \end{bmatrix}\]

• “Robust” standard errors using the Eicker-Huber-White “sandwich” estimator - Consistent but not unbiased for the true sampling variance of \(\hat{\beta}\)

  \[\widehat{Var(\hat{\beta})} = (\mathbf{X}^{\prime}\mathbf{X})^{-1} \mathbf{X}^{\prime}\hat{\Sigma}\mathbf{X}(\mathbf{X}^{\prime}\mathbf{X})^{-1}\]

  • \(\hat{\Sigma}\) is our estimate of the variance-covariance matrix using the squared residuals on the diagonals
  • Extensions to “clustered” standard errors that allow arbitrary correlation within groups.

• What do we need for OLS to be consistent for the “best linear approximation” to the CEF?
  • Very little!
• What do we need for OLS to be consistent and unbiased for the conditional expectation function?
  • Truly linear CEF
  • But still no assumptions about the outcome distribution!
• What do we need to do inference on \(\hat{\beta}\)?
  • We almost never assume homoskedasticity because “robust” SE estimators are ubiquitous
  • Even some forms of error correlation are permitted (“cluster” robust SEs)
  • Sample sizes are usually large enough where Central Limit Theorem implies a normal sampling distribution is a reasonable approximation.